Copyright 1996,1997, All rights reserved, Izumi Ohzawa,

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1 CXmodelWhyBB-BD.ma 1 This Mathematica Notebook contains computations and derivations used to create plots in Figs. 8, 9 and 10 of our paper Ohzawa, DeAngelis, Freeman, Encoding of Binocular Disparity by Complex Cells in the Cat s Visual Cortex, J. Neurophysiol. 77: , (1997). Copyright 1996,1997, All rights reserved, Izumi Ohzawa, izumi@pinoko.berkeley.edu Monocular response (Left eye) of Cx cell based on the energy model is Lresp (xl) = (Exp[-k xl^2] Cos[2 Pi xl])^2 + (Exp[-k xl^2] Sin[2 Pi xl])^2 = (Exp[-k xl^2])^2 = Exp[-2 k xl^2] k is proportional to the inverse of subunit RF width, and pd is the phase difference of L and R subunit RF fields (same for all 4 subunits). Monocular response for the Left eye is given by,

2 CXmodelWhyBB-BD.ma 2 z[x_]:=graylevel[1-x]; pd:= 0; Exp[- 2 k xl^2], Contours -> 12, PlotRange -> {0, 2.2}, Sum of Left and Right monocular responses (which may be a good model for a non-phase specific cells) is given by:

3 CXmodelWhyBB-BD.ma 3 z[x_]:=graylevel[1-x]; pd:= 0; ( Exp[- 2 k xl^2] + Exp[-2 k xr^2] ), Contours -> 12, PlotRange -> {0, 2.2}, This is the derivation of Equation 7 from Equation 5 in our paper Ohzawa, DeAngelis, Freeman, Encoding of Binocular Disparity by Complex Cells in the Cat s Visual Cortex, J. Neurophysiol. 77: , (1997). Binocular response for matched contrast conditions (BB and DD) may be simplified as follows:

4 CXmodelWhyBB-BD.ma 4 Simplify[ ( Exp[-k xl^2] Cos[2 Pi F xl] +Exp[-k xr^2] Cos[2 Pi F xr + pd])^2 +(Exp[-k xl^2] Sin[2 Pi F xl] +Exp[-k xr^2] Sin[2 Pi F xr + pd])^2 ] k xl -2 k xr E + E + 2 Cos[pD - 2 F Pi xl + 2 F Pi xr] k (xl + xr ) E Just to be general, try the case where spatial freq are different between L and R eyes (use FL and FR for each eye). Simplify[ (Exp[-k xl^2] Cos[2 Pi FL xl] + Exp[-k xr^2] Cos[2 Pi FR xr + pd])^2 +(Exp[-k xl^2] Sin[2 Pi FL xl] + Exp[-k xr^2] Sin[2 Pi FR xr + pd])^2 ] k xl -2 k xr E + E + 2 Cos[pD - 2 FL Pi xl + 2 FR Pi xr] k (xl + xr ) E As shown above, the first two terms are monocular responses. The last term is the binocular interaction component, which is a Gabor function (Is this really a Gabor?) that is oriented at 45 degs in (xl, xr) domain (see below). The binocular component alone may be obtained without separate monoculuar measurements by computing BB+DD-BD-DB. Since BB == DD and BD == DB, because of squaring which makes the function independent of the inversion of sign, BB+DD-BD-DB = 2 (BB-BD) BB-BD is given by below which is reduced the same expression as above.

5 CXmodelWhyBB-BD.ma 5 k=.; pd=.; Simplify[ (Exp[-k xl^2] Cos[2 Pi xl] +Exp[-k (xr)^2] Cos[2 Pi (xr) + pd])^2 +(Exp[-k xl^2] Sin[2 Pi xl] +Exp[-k (xr)^2] Sin[2 Pi (xr) + pd])^2 - ( (Exp[-k xl^2] Cos[2 Pi xl] -Exp[-k (xr)^2] Cos[2 Pi (xr) + pd])^2 +(Exp[-k xl^2] Sin[2 Pi xl] -Exp[-k (xr)^2] Sin[2 Pi (xr) + pd])^2 ) ] 4 Cos[pD - 2 Pi xl + 2 Pi xr] k (xl + xr ) E This is a Gabor function oriented at 45 degs in the (xl, xr) domain as shown below: (ModelDecomp/GaborLR-0.eps)

6 CXmodelWhyBB-BD.ma 6 z[x_]:=graylevel[1-2*abs[x-0.5]]; pd:= 0; Exp[-k (xl^2 + xr^2)] Cos[2 Pi (xl - xr) - pd], Contours -> 12, PlotRange -> {-1.1, 1.1},

7 CXmodelWhyBB-BD.ma 7 Plot[ Exp[-k (xl^2)] Cos[2 Pi xl], {xl, -1, 1}, PlotPoints -> 40 ] Graphics- Phase Difference = 45 degs (ModelDecomp/GaborLR-45.eps)

8 CXmodelWhyBB-BD.ma 8 z[x_]:=graylevel[1-2*abs[x-0.5]]; pd:= Pi/4; Exp[-k (xl^2 + xr^2)] Cos[2 Pi (xl - xr) - pd], Contours -> 12, PlotRange -> {-1.1, 1.1}, Phase Difference = 90 degs (ModelDecomp/GaborLR-90.eps)

9 CXmodelWhyBB-BD.ma 9 z[x_]:=graylevel[1-2*abs[x-0.5]]; pd:= Pi/2; Exp[-k (xl^2 + xr^2)] Cos[2 Pi (xl - xr) - pd], Contours -> 12, PlotRange -> {-1.1, 1.1}, Phase Difference = 180 degs (ModelDecomp/GaborLR-180.eps)

10 CXmodelWhyBB-BD.ma 10 z[x_]:=graylevel[1-2*abs[x-0.5]]; pd:= Pi; Exp[-k (xl^2 + xr^2)] Cos[2 Pi (xl - xr) - pd], Contours -> 12, PlotRange -> {-1.1, 1.1}, Zero disparity detector for both phase and position models is the same:

11 CXmodelWhyBB-BD.ma 11 z[x_]:=graylevel[1-x]; pd:= 0; Exp[- 2 k xl^2] + Exp[-2 k xr^2] + 2 Exp[-k (xl^2 + xr^2)] Cos[2 Pi (xl - xr) - pd], Contours -> 12, PlotRange -> {0, 4.4}, Non-Zero disparity detector for the phase model

12 CXmodelWhyBB-BD.ma 12 z[x_]:=graylevel[1-x]; pd:= Pi/2; Exp[- 2 k xl^2] + Exp[-2 k xr^2] + 2 Exp[-k (xl^2 + xr^2)] Cos[2 Pi (xl - xr) - pd], Contours -> 12, PlotRange -> {0, 4.4}, Non-Zero disparity dector for the Position model

13 CXmodelWhyBB-BD.ma 13 z[x_]:=graylevel[1-x]; or:= 0.25; pd:= 0; Exp[- 2 k xl^2] + Exp[-2 k (xr+or)^2] + 2 Exp[-k (xl^2 + (xr+or)^2)] Cos[2 Pi (xl - (xr+or)) - pd], Contours -> 12, PlotRange -> {0, 4.4},

14 CXmodelWhyBB-BD.ma 14 z[x_]:=graylevel[1-x]; pd:= Pi/2; Exp[-k (xl^2 + xr^2)] Cos[2 Pi (xl - xr) - pd], Contours -> 12, PlotRange -> {-1.1, 1.1},